Uniqueness of cubic interpolating polynomial

This is a numerical analysis question, and I am trying to prove that the p(0), p'(0), p(1), p'(1) define a unique cubic polynomial, p. More precisely, given four real numbers, p00, p01, p10, p11, there is one and only one polynomial, p, of degree at most 3 such that p(0) = p00, p'(0) = p01, p(1) = p10, p'(1) = p11.

I looked in a lot of numerical analysis textbooks, but the closest proof I can find proves the following: Given n+1 distinct points x0,..., xn and n + 1 values y0, ..., yn, there exists a unique polynomial p of degree of most n with the property that p(xj) = yj, for j = 0,...., n.

For my problem I have no idea what I should do with p(0), p'(0), p(1), p'(1), the values and derivatives of p at 0 and 1. Thanks.

Ok that makes sense. So I guess what you are using is the Fundamental Theorem of Linear Algebra -- determinant of the coefficients is not 0 iff the coefficients of the polynomial are unique? Thanks for your help.